SCA 2001-35
MODELING OF CAPILLARY PRESSURE FOR HETEROGENEOUS
RESERVOIRS BY A MODIFIED J-FUNCTION
1
M.Sarwaruddin 1, A. Skauge2 and O. Torsæter1
Norwegian University of Science & Technology, 2Norsk Hydro
ABSTRACT
Initial fluid distribution for heterogeneous reservoirs i.e. layered reservoirs with different permeability
and porosity is often modeled by simplified capillary J-function for input to reservoir simulation.
Leverett introduced the model in 1941 and showed that dimensionless capillary J (Sw)-function varies
only with fluid saturation. However, theories of capillary pressure indicate that Leverett J-function
should also be a function of tortuosity, irreducible liquid saturation and pore size distribution. As a
result, scaling of capillary pressure by Leverett J-function should be limited to the formations whose
pore size distribution and irreducible liquid saturation are equal.
The pore size distribution is likely to affect the shape of the drainage capillary pressure (Pc) saturation curve. Therefore if pore size distribution is different in two formations, saturation exponent,
a key parameter for the drainage Pc curve is likely to be different. The effect of tortuosity on Jfunction is not pronounced since its value does not change too much. However, if Leverett J-function
is used to obtain Pc-saturation functions for different formations whose irreducible liquid saturation
and the saturation exponent are different, then significant saturation discrepancy might arise in the
scaled Pc-functions.
It is also observed that irreducible liquid saturation is a function of specific surface area, wettability
and interfacial tension of the rock-fluid system. Therefore scaling of Pc by Leverett J-function may
provide incorrect Pc-saturation value at low saturation range even for the similar rocks but whose
fluid systems are different. In this paper, we define a modified J-function that includes tortuosity,
irreducible liquid saturation and saturation exponent so that the modified J-function may be used for
scaling of Pc-saturation functions for any formations. We also propose auxiliary equations for
estimating tortuosity, irreducible liquid saturation and saturation exponent from permeability and
porosity data that may be used to calculate the modified J-function.
INTRODUCTION
Fluid distributions in heterogeneous reservoirs depend on many factors. Lee1 discussed the
importance of temperature, fluid composition, phase pressures, geothermal gradient, fault, rock-fluid
wettability as well as history of fluid trapping as potential elements for distributions of fluid in
reservoirs. Alpak et al2 theoretically showed how the capillary pressure might vary in porous media
due to different interfacial area arises out of fluid saturation and pore structure. However, for all
practical purpose it is assumed that drainage capillary pressure may define the initial fluid saturation
for a given rock-fluids system. The variations of rock type and fluid system however, generate
different capillary pressure saturation functions.
1
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SCA 2001-35
In 1941, Leverett 3 proposed a dimensionless capillary J-function with an intention to eliminate the
effect of rock heterogeneity i.e. effect of porosity and permeability variations from the capillary
pressure-saturation function. Since then it is widely used as a scaling function for capillary pressure
modeling. However, Melrose4 found that J-function could not scale capillary pressure at lower
saturation typically below 20 percent even for Berea cores that are usually assumed homogeneous.
Phelps5 also found lithology dependent J-function for heterogeneous reservoirs. El-khatib 6 showed
that Leverett J-function is not only a function of wetting fluid saturation but also a function of
tortuosity, saturation exponent (a parameter for drainage capillary pressure related to pore size
distribution of the porous media) and irreducible wetting fluid saturation. Lake7 also pointed that
variation of pore size distribution will affect the J-function.
THE THEORY OF J-FUNCTION (LITERATURE REVIEW)
In 1941, Leverett investigated the effect of fluids interfacial tension (σ) and rock properties i.e.
permeability (k) and porosity (φ) on capillary pressure. Based on experiments, he proposed following
dimensionless group as a function of wetting fluid saturation (Sw) for capillary pressure (P c) modeling
J (S w ) ≡
Pc
σ Cosθ
k
= f (S w ) L (1)
φ
Here, θ = contact angle, a property of rock-fluid system. f (Sw)= a saturation function.
However, it has been reported in the literature 1,2,5 that J-function often failed to scale Pc correctly for
heterogeneous reservoirs. The possible reasons may be attributed to complex lithology, and the
variation of fluid composition in the reservoirs.
Using simplified porous model such as bundle of capillary tubes, fundamental fluid flow equation for
capillary tube (Hagen-Poiseuille) and Young-Laplace equation for capillary pressure, El-khatib 6
showed that permeability (k) may be expressed in terms of porosity (φ), tortuosity (τ), contact angle
(θ), wetting fluid saturation (Sw) and Pc.
k=
φ
(σ Cosθ )2
2τ
1
dS w
0
Pc2
∫
L (2 )
For analytical purpose, we will assume that the following form of saturation function can model the
drainage Pc. The sole purpose of assuming a Pc function is to show that Eq.1 is not a function of
saturation alone. Therefore any other form of Pc-saturation function will reveal the same.
Pc = ∞,0 ≤ S w ≤ S wir ; Pc =
a
 S w − S wir

 1 − S wir



n
S wir < S w ≤ 1 K (3)
Here, a, n and Swir are matching parameters for a Pc-saturation function. However, each of these
parameters indicates some properties of the porous media. For example parameter "a" indicates the
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SCA 2001-35
minimum required capillary entry pressure that in turn provide the maximum pore size of the given
rock. Swir, the irreducible wetting fluid saturation indicates the specific surface area of the given rock.
The exponent "n" indicates the pore size distribution pattern. Now, substituting Eq.3 in Eq.2, one can
express permeability (k) as:
k=
φ
(σ cosθ )2 2 1 (1 − S wi ) K(4)
2τ
a (2n + 1)
Dividing Eq.3 by Eq.4 and using the identity of Eq.1, J-function can be expressed:
J=
1 − S wir
1
2τ .(2n + 1)  S w − S wir

 1− S
wir





K (5)
n
Now if a modified J-function is defined such that J* equal to,
J * = J.
2τ .(2n + 1)
1 − S wir
K (6)
Then J* may be written in normalized saturation, Sw*≡(Sw-Swir)/(1-Swir) as:
J *=
1
(S )
* n
w
K (7)
Eq.7 shows that J* is a function of S w* and the saturation exponent "n". Now, if we know "τ", "n" and
Swir, the scaling of Pc may be accomplished using Eq.1 and Eq.6. In the following section we would
discuss a solution procedure for Eq.7.
SOLUTION PROCEDURES FOR MODIFIED J-FUNCTION
The objective of the modified J-function is to take into account the effect of tortuosity, irreducible
liquid saturation and pore size distribution when Pc-saturation function is scaled from one formation
to another. However, the unknown Pc-saturation function may be scaled only if k,φ and (σ.cosθ) are
known in both formations. Note that in order to calculate J* for a formation whose Pc-saturation
function is known, one do not need to calculate τ, n and Swir because n and Swir can be estimated from
Pc -function (Eq.3) and the right hand side of Eq.7 may then be used to calculate J*. Using different
n-values, family of J* vs. Sw* can be plotted (see Fig.1), which may then be used to equate J* if n can
be estimated for the unknown system. If Swir can also be estimated for the unknown system, the actual
saturation may be calculated from the following:
1/ n
S w = S wir
 1 
+ (1 − S wir ). * 
J 
3
K (8)
SCA 2001-35
Now, one can solve Eq.6 for Pc if τ can also be estimated for the unknown system. By doing the
above, one can obtain a pair of (Pc, Sw) for the unknown system. The calculation process can be
repeated with different values of J* in order to obtain full Pc-saturation function for the unknown
system. In the following section we would discuss how one may estimate n, Swir and τ.
Irreducible water saturation (Swir)
Specific surface area (SV) of the porous media is expected to be directly proportional to the irreducible
water saturation. The SV, which is defined as total surface area per unit flow volume, may be related
to the characteristic size (CS) and the porosity (φ) of the packing. The CS may be defined as the ratio
of volume to surface area of a single grain, for example the CS equals to DP/6 for a sphere of size DP.
If spheres of regular size constitute the porous media, the SV may then be written for this type of
packing as:
S V = (1 − φ ) / φ C S
L (9)
In real porous media, the SV is difficult to calculate due to the presence of different sizes of grains.
However, an equivalent constant CL that may be considered as characteristic length for a particular
porous media may be used to replace CS in Eq.9. The irreducible water saturation Swir may then be
written as follows since SV is proportional to Swir.
S wir = (1 − φ ) / φ C L CW
L (10 )
Where, CW is a proportional constant and may be calculated dividing SV by Swir. The term (1φ)/(CL.CW) in Eq.10 may be replaced by another constant CR that depends on the sedimentation
process and the characteristic length of the porous media. In other words, CR will depend on the
permeability of the porous media. Considering the above arguments, S wir may be written as:
S wir =
CR
φ
K (11)
Based on experimental observations from their own and from the other published sources, Sanyal12
and Homes13 et al. also concluded the above relationship for Swir. One apparent problem of Eq.11
comes from the fact that for many reservoirs, variation of φ is insignificant compare to S wir. Therefore,
in order to estimate Swir from Eq.11, CR must be estimated very accurately for different reservoir
zones. Intuitively one may consider that value of CR will increase if specific surface area SV increased.
In other words, if specific surface area changes, permeability is likely to change and CR may then be
estimated as a function of permeability:
C R ≡ S wir .φ = f (k ) K(12)
Where, f(k) is a function of permeability. In order to obtain f(k), we plot (S wir.φ) vs. k (see Fig.2). The
most of the data were taken from literature9-11 and the rest were taken from our own experiments. Note
that we considered only air/N2-water system and did not plot any data from carbonate samples. A
hyperbolic decline function shown below was fitted to data (see Fig.2).
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SCA 2001-35
f (k ) =
α .β
β +k
K (13)
Where, α = constant, regressed value = 0.1683, β = constant, regressed value =96.93
Now, Swir may be calculated from Eq.12 replacing CR = f(k).
Irreducible liquid saturation vs. Interfacial tension (σ)
To address the effect of interfacial tension on irreducible liquid saturation, we conducted experiments
on four Berea samples (B1, B2, B3 and B12). First the water-saturated samples were centrifuged to
determine Swir then Pc-saturation functions were determined by directly monitoring the saturation 14,15
for the same samples using a different fluid system (Air-C19). From the Pc-saturation curve, we
determine irreducible liquid (C19) saturation for each sample. The irreducible liquid (C19) saturation
was scaled by a factor = σ(air-water: 72 mN/m)/ σ(air- C19 : 22.3 mN/m) and then it was plotted
against the corresponding Swir (see Fig.3). We found almost a diagonal relationship between the two.
Therefore, we conclude irreducible liquid saturation for one fluid system may be scaled from another
fluid system by the ratio of interfacial tension.
Tortuosity (τ)
The tortuosity (τ) may be expressed in terms of the product of porosity and formation factor 11, 16.
Formation factor F may be defined as ratio of electrical resistivity of water saturated porous medium
(Ro) to that of the water alone (Rw). Thus τ may be written as:
τ=
Ro
.φ = F .φ K (14)
Rw
The development of Eq.14 is based on simplified tortuous path; therefore it is expected that Eq.14
needs modification for its application to the porous media. Amyx11 et al reported a correction for
Eq.14 based on experimental works of others, which may be written as:
τ = (F .φ )1.2
K (15)
There are several electrical resistivity models reported in the literature17 for various formations type.
In order to calculate τ from Eq.15, one can either directly measure F or select a model depending on
the rock type. For simplicity, we would consider Archie's resistivity model18 so that F may be written
as:
F = c.φ − m
K (16 )
Where, m= cementation exponent and c= constant. The value of c is considered to be 1 from the fact
that when φ=1,then F=1,therefore c=1. However, a lower value of "c"(say 0.81) is also reported for
some formation8. Thus, considering c=1, τ may be expressed:
τ = φ −1.2.(m −1) L(17 )
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Dullien 16 and Sanyal12 reported a range (1.3 ≤ m ≤ 2.5) for "m" therefore, one can calculate τ from
Eq.17 if cementing property of the formation can be estimated qualitatively. For an approximate
estimation of "m", Fig.4 was constructed based on the experimental "m" values11. Note that cementing
ranking of the formation is arbitrary i.e. we refer reservoir rank1 unconsolidated while rank10 may be
considered highly cemented.
Saturation exponent "n"
For a given lithology, the product of irreducible liquid saturation (S wir) and the saturation exponent
“n” may be expressed as function of permeability (k) and porosity (φ). At a given k and φ, the effect
of interfacial tension on "n.S wir" is likely to be insignificant since it is expected that when irreducible
liquid saturation increases, the saturation exponent "n" decreases proportionally.
In order to determine "n.Swir" with respect to k and φ, we conducted experiments in macro and micro
scale to obtain necessary data. In macro scale, Pc-saturation functions (oil-brine, σ:36 mN/m) were
determined for 11 samples (v2k/1-v2k/11; see Table1 for basic data) by porous plate method. The
samples were taken from various depth of a North Sea field. In micro scale, Pc-saturation functions
(C19-air, σ:22.3 mN/m) were determined by saturation monitoring15 for three more samples (v128,
v129 and v130) from the same North Sea field but from different formations (see Table2 for basic
data). Some Pc data from various fluid systems were also taken from literature 8-11,14 to validate the
assumption that effect of interfacial tension is insignificant with respect to n.Swir vs. k, φ relation.
The porous plate and reported literature data are plotted in Fig5-6 while micro scale experimental data
are plotted in Fig7-9. Note that n.Swir are plotted against the so called reservoir quality index i.e.
(k/φ)0.5 . An exponential decay function of the following form was fitted to each group of data.
n.S wir = λ .e
k
−γ  
φ 
0. 5
L (18 )
Note that the parameters λ and γ (Eq.18) are not constant (Fig5-9). Therefore, we suggest when Pc
scaling is conducted in accordance with Eq.6-7, at least couple of full Pc-saturation functions are
determined so that λ and γ may be estimated satisfying the known pair of "n, S wir, k and φ ".
EXPERIMENTAL PROCEDURES: MICRO SCALE
Very fine (mm) scale porosity (φ) was measured (see Fig.10) for samples v128, v129 and v130 by
gamma absorption (see Appendix: Micro scale porosity). Initially an attempt was also made to
measure permeability distribution along the length of the samples by a mini permeability meter 19.
Failing to design an appropriate air injection nozzle, we could not use the instrument for permeability
measurement in mm scale. However, the mm scale permeability (k) was determined from φ-k
correlation obtained by collecting large number of field porosity-permeability data (see Fig.11).
Capillary pressure-saturation functions were determined for these samples in two different methodsHassler-Brunner20 and saturation monitoring 14,15. Note that in the Hassler-Brunner method, each
sample was cut into two pieces (1"x1" approximately) before they were put into the centrifuge
(Beckman L8-55M/P). The data set {centrifuge speed, average liquid saturation} from each piece
(total six; each sample consisting two pieces) was interpreted with radial correction21.
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SCA 2001-35
From Hassler-Brunner type experiment, average irreducible liquid saturation and average saturation
exponent "n" was determined for each sample by regression (see Fig.12a, 13a and 14a) while effect of
mm scale heterogeneity (k-φ variations) on the Pc-saturation functions were captured by saturation
monitoring method (see Fig.12b, 13b and 14b). For details of saturation monitoring method see
Appendix: Capillary pressure-saturation function by saturation monitoring. From Fig.12a-b, 13a-b
and 14a-b, it can be seen that at a given Pc value, there is significant difference in saturation values
between the Hassler-Brunner and saturation monitoring methods. The reason for these differences
may be attributed to the local heterogeneity in mm scale.
For each of the Pc-saturation functions shown in Fig12b-14b, Eq.7 was solved for saturation exponent
"n" using the local Pc, k, φ, Swir, τ and σ data. Note that some local Pc data points were discarded due
to no solution for Eq.7.
CONCLUSIONS
1. The limitation of Leverett J-function is investigated and it has been shown that Leverett Jfunction is suitable for capillary pressure-saturation (Pc-Sw) function scaling from one formation
to another if lithology (i.e. pore size distribution, irreducible liquid saturation and tortuosity) of
the two formations is same.
2. To overcome the limitation of Leverett J-function, a modified J-function is defined which
includes pore size distribution effect in the form of saturation exponent, irreducible liquid
saturation and tortuosity.
3. Solution procedures for estimating saturation exponent, irreducible liquid saturation and tortuosity
have been discussed those may be used to calculate modified J-function.
4. When lithology of the two formations are different, scaling of Pc-function by modified J-function
is expected to provide reliable saturation values for the scaled Pc-function than the Leverett Jfunction.
Pc = Capillary Pressure (kPa)
R = Resitivity (ohm), subscript -o: Formation, w=
water
S = Saturation, subscript-w: wetting phase, wir
=wetting phase irreducible
α= Constant, β= constant (Eq.16)
σ= Interfacial tension (mN/m)
θ = Contact angle
τ = Tortuosity
φ = Porosity
NOMENCLATURE
a = Capillary entry pressure (kPa), constant for a
given system (Eq.3)
c =Constant (Eq.12)
CR = a function of permeability (md) (Eq.15)
F= Formation factor (Eq.10)
J= Leverett J-function (Eq.1)
J* = Modified J-function (Eq.6)
k = Permeability (md)
m = Cementation exponent (Eq.12)
n = Saturation exponent (Eq.3)
7
SCA 2001-35
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
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presented at the SPE 64th Annual Technical Conference, San Antonio, Tx, Oct 8-11,1989.
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Relative permeabilities," paper SPE presented at the SPE Annual Technical Conference,1999, Houston, Tx.
Oct. 3-6,1999
Leverett, M.C.:" Capillary Behavior in Porous Solids," Trans. AIME (1941) 42, 152-169.
Melrose,J.C.:" Scaling Procedures for Capillary Pressure Data at Low Wetting-Phase saturation," Paper
SPE 19594 presented at the SPE 64th Annual Technical Conference, San Antonio, Tx, Oct 8-11,1989.
Phelps, R.E.:" Lithology-Dependent J-Functions and Relative Permeabilities," Paper SPE 25661 Presented
at the SPE Middle East Oil Technical Conference, Bahrain, April 3-6, 1993.
El-Khatib,N.:" Development of a Modified Capillary Pressure J-Function," Paper SPE 29890 presented at
the SPE Middle East Oil Show in Bahrain, March 11-14, 1995.
Lake, W.L.:" Enhance Oil Recovery," Prentice-Hall, Inc. NJ 07632, (1989), pp. 54-56.
Baldwin,B.A. and Yamanashi, W.S., " Capillary-Pressure determinations from NMR Images of centrifuged
Core Plugs: Berea sandstone," The Log Analyst (Sept.-Oct. 1991 pp. 550-556
Hermansen, H., Eliassen, Ø., Guo,Y., and Skjæveland, S.M.:" Capillary Pressure From Centrifuge- A New
direct Method," Advances in Core Evaluation II, EuroCAS (1991)
Hustad, O.S. and Holt,T.:" Gravity stable Displacement of Oil by Hydrocarbon Gas After waterflooding,"
Paper SPE/DOE 24116 presented at the Eight Symposium on Enhanced Oil Recovery, Tulsa, Oklahoma,
April 22-24,1992.
Amyx, W.J., Brass-Jr, M.D. and Whiting, L.R.:" Petroleum Reservoir Engineering," McGraw-Hill, Inc,
Classic Text Book Reissue Series (1988), pp. 110-117, 148-149.
Sanyal, S.K. and Ellithrope, J.E.:" A Generalized Resistivity-Porosity Crossplot Concept," paper SPE 7145
presented at the Cailfornia Regional Meeting of the SPE of AIME, San Francisco, California, April 12-14,
1978.
Homes, M., Dreher, K.D. and Eloy, A.J.:" Lithology and Fluid Properties and Their Relations to the Fluids
in Sandstones," Paper SPE 3554 presented at the SPE 46th Annual Fall Meeting, New Orleans, LA, Oct. 36, 1971.
Spinler, E.A. and Baldwin, B.A.:" Capillary Pressure Scanning Curves By Direct Measurement of
Saturation," Proceedings of the 1997 SCA International Symposium,1997
15. Sarwaruddin, M., Torsæter, O. and Skauge, A.:" Comparing Different Methods for Capillary pressure Measurements",
SCA 2000-51, 2000, Abu Dhabi, UAE.
16. Dullien, F.A.L.:" Porous Media Fluid Transport and Pore structure," Academic Press, Inc. San Diego, California 2nd
ed.1992, pp. 95, 306-308
17. Jorden, R.J. and Campbell, F.L.:" Well Logging II-Electric and Acoustic Logging", SPE Monograph
Vol.10, New York, 1986 pp. 26-27.
18. Archie, G.E.:" Electrical Resistivity Log as an Aid in Determining Some Reservoir Characteristics," Trans.,
AIME (1942) 146, pp. 54-62.
19. Goggin, D.J, Thrasher, R. and Lake, L.W.:" Theoretical Analysis of Minipermeameter Response," 3 rd
Annual CEOGRR report, March 1986, category A research, University of Texas, Austin, pp 148-167.
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(1945) 160, pp.114-123.
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Pressure Curves," Paper SPE 28182 presented at the 69th SPE Annual Conference in New Orleans, LA, 2528 Sep. 1994.
22. Sarwaruddin, M. " Modeling of Capillary Pressure Hysteresis by Saturation Monitoring," Ph.D. thesis, The
Norwegian University of Science & Technology, Trondheim, Norway, in progress (2001).
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Paper SPE 14421, SPE Formation Evaluation, March 1988, pp.69-75
24. Knoll, G. F.:" Radiation Detection and Measurement", John Wiley & Sons (1989) Second ed., pp. 80-88
8
SCA 2001-35
Table-1: Basic data for v2k/1-v2k/11
φ
188
259
41.2
1630
1220
1710
0.532
2201
33.9
840
44.8
100
0.246
0.258
0.217
0.262
0.256
0.262
0.154
0.287
0.211
0.239
0.246
n=0.5
n=1
J*
k
10
n=1.5
n=2
1
0
0.2
0.4
0.6
0.8
1
Sw*
Figure1: Modified J-function vs. normalized
saturation for different n-values
Table-2: Basic data for v128-v130
φ
k
0.2
25.2
25.6
25.3
0.15
Swi.'Ø
16
35
18
0.05
Air
0.1
0
0.05
fluid
)
0.15
/
Sw i .Ø
0.1
Swi *(
0.2
CR
0
1000 2000 3000
k (md)
0
0
0.05
0.1
0.15
0.2
Swi (Air-water)
Figure2: Irreducible water saturation vs. permeability. Data is
taken from literature9-11 and from our own experiments. A
hyperbolic function was fit to data
2.4
2.2
n.Swir
Cementation exponent "m"
Figure3: Effect of interfacial tension on irreducible liquid saturation.
2
1.8
1.6
Cementing Rank:
1: unconsolidated
2~3: very slightly cementated
4~5: slightly cementated
6~7: Moderately cementated
8~10: Highly cmentated
1.4
1.2
1
0
5
10
0
15
20
40
60
80
(k/?) 0.5
Formation cementing rank
Figure4: Cementation exponent "m" is plotted
against formation cementing Rank11
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure5: n.Swir vs. (k/φ) 0.5 for samples v2k/1-v2k11
9
100
0.8
1
0.6
0.8
n.Swi
n.Swi
SCA 2001-35
0.4
0.2
0.6
v128
0.4
best-fit
0.2
0
0
0
20
40
60
80 100 120
0
20
0.5
Figure6: n.Swir vs. (k/φ) 0.5 from literature8-11
0.3
0.25
0.2
0.15
0.1
0.05
0
0.3
v129
0.2
best-fit
n.Swi
n.swi
0.4
0.1
0
40
60
80
Figure7: n.Swir vs. (k/φ) 0.5 from micro scale data: sample v128
0.5
20
60
(k/?)0.5
(k/??
0
40
80
v130
best-fit
0
20
40
60
80
0.5
(k/?)
(k/?)0.5
Figure8: n.Swir vs. (k/φ) 0.5 from micro scale data: sample
v129
Figure9: n.Swir vs. (k/φ) 0.5 from micro scale data: sample
v130
5
0.3
4
log(k) = 29.515φ - 5.4267
R2 = 0.6857
log(k), k(md)
Porosity (fraction)
3
0.25
v128
v129
v130
0.2
2
1
0
-1
-2
-3
0.15
0
0
20
40
60
0.1
0.2
0.3
Porosity (fraction)
Position (mm)
Figure11: Field permeability-porosity (k-φ) correlation
Figure10: variation of local porosity for the
heterogeneous samples
10
0.4
SCA 2001-35
200
200
n=1.246
Swir=0.175
150
v128
100
Pc(A)
Pc (kPa)
Pc (kPa)
150
v128
100
Pc(A)
50
50
0
0
0
0
0.2
0.4
0.6
0.8
0.2
0.4
1
0.6
0.8
1
Sw
Sw
Figure12b: Capillary pressure is determined by saturation
monitoring15 method for sample-v128. Pc (A) is the analytical Pc
that is redrawn from Fig.12-a. Significant saturation discrepancy
is observed between these two methods for this heterogeneous
sample.
Figure12a: Capillary pressure is determined by HasslerBrunner20 method for sample-v128. Pc (A) is the best-fit
analytical Pc function.
160
140
Pc (kPa)
120
100
v129
80
Pc(A)
60
40
20
0
0
0.2
0.4
0.6
0.8
Pc (kPa)
n=1.3
Swir=0.16
200
180
160
140
120
100
80
60
40
20
0
v129
Pc(A)
0
1
0.2
0.4
0.6
0.8
1
Sw
Sw
Figure13a: Capillary pressure is determined by Hassler-Brunner20
method for sample-v129. Pc (A) is the best-fit analytical Pc function.
11
Figure13b: Capillary pressure is determined by saturation
monitoring15 method for sample-v129. Pc (A) is the
analytical Pc that is redrawn from Fig.13-a. Saturation
discrepancy is however not observed like the sample v128.
180
160
140
120
100
80
60
40
20
0
n=1.71
Swir=0.15
v130
Pc (kPa)
Pc (kPa)
SCA 2001-35
Pc(A)
0
0.2
0.4
0.6
0.8
1
180
160
140
120
100
80
60
40
20
0
v130
Pc(A)
0
Sw
0.2
0.4
0.6
0.8
1
Sw
Figure14a: Capillary pressure is determined by HasslerBrunner20 method for sample-v130. Pc (A) is the best-fit
analytical function.
Figure14b: Capillary pressure is determined by saturation
monitoring15 method for sample-v130. Pc (A) is the
analytical Pc that is redrawn from Fig.13-a. Saturation
discrepancy is however unlike the sample v128.
APPENDIX
Micro scale porosity
Gamma-ray
detector
241
A radioactive scanner (source: Am, half-life:
433 years, energy 59.5 KeV; detector: NaI-Tl)
was used to capture the core scale porosity
variation in millimeter scale (see Fig.A1 for set
up of the instrument). The governing equation
for estimating porosity (φ) at any location of
the sample can be written by the following
equation:
I C = I *e
[
− (1−φ )µ s +φµg + µ B
]
Source-Detector
movement
syncronized by
computer.
L ( A − 1)
Where, IC = Number of gamma photons
counted in the detector per unit time, I* =
Number of incident gamma photons per unit
time. µ s, µ g and µ B = absorption coefficients
for matrix (solid), gas (air) and the background.
Note that µ s, µ g are not defined as linear
attenuation coefficients rather they may be
regarded as lumped attenuation coefficients
whose magnitude depend on the samples'
geometry, orientation and its position within
source and detector (see Fig.A-1).
Gamma-ray
source
FigureA-1: Schematic of saturation
monitoring by γ-absorption
In order to determine porosity (φ) from Eq.A-1,
it is necessary to know the absorption
coefficients (µ s, µ g) and the intensity of the
gamma source. The I* may be written in terms
12
SCA 2001-35
instrument counts only a fan of attenuated γrays from a thin x-section of a sample. For a
given location j, the recorded count rate, IC,j
may then be written like the Eq.A-1.
of background scanning counts (IB) i.e. without
placing any sample in between source and
detector.
I B = I *e
(
− µ B +µ g
)
K ( A − 2)
I C, j = I j e
Dividing Eq.A-2 by Eq.A-1 and rearranging
the terms, one can write following:
I
ln B
 IC
)
After finishing the radioactive scanning all
along the dried samples, it is required that the
samples are saturated fully. The samples are
then scanned again exactly on the same
locations (j =1, N) like the dry samples. At
location j, the recorded count rate, IS,j for
saturated samples may then be written as
For a given cylindrical rock, µ s and µ g are
constant. The difference of the coefficients i.e.
(µ s-µ g) may be determined if we take few dry
samples so that experimental values of
ln(IB/IC,avg) vs. φavg may be fitted to a linear
function that passes through origin. The slope
provides the value (µs-µ g) while the local
porosity variations for similar samples may
then be calculated by Eq.A-3 if value of IC is
known at that location. Here, the term IC,avg is
defined as
1
N
L ( A − 4)
Where, Ij = I*.e-(µB+µs(1-φ)j). φj = porosity at
location j.

 = µ s − µ g .(1 − φ ) K ( A − 3)


(
− µ gφ j
IS, j = I je
− µ lφ j
L ( A − 5)
Where, µ l = γ-ray absorption coefficient for
saturating liquid.
N
∑I
C, j
, where, j = an index
For capillary pressure measurement, the
saturated samples are then rotated at a fixed
speed so that a saturation profile is established
within the sample due to equilibrium between
capillary and centrifugal force. If we choose a
saturating liquid whose melting point is higher
than the room temperature then the equilibrium
saturation profile can be locked lowering the
system temperature to room temperature. The
samples may then be scanned like the other
two scans and for location j, we can write
recorded count rate, IT,j :
j =1
assigned to identify specific location (xsection) of a given sample, N = maximum
number of locations where radioactive
scanning is performed. φavg is defined as
sample's average porosity measured by helium
porosity meter.
Capillary pressure-saturation function by
saturation monitoring
Capillary pressure and fluid saturation
relationship for reservoir cores may be
determined by saturation monitoring. The
method requires that the samples are cleaned
and dried. The samples may then be wrapped
very tightly for example by heat-shrinking
sleeve so that only end faces remain open for
flow. At this stage the samples are placed in
between a γ-ray source and a detector (see
Fig.A1). The γ-source and detector slots
(openings) are set in such a way that the
IT , j = I j e
(
− µ l φ j S o , j + µ gφ j S g , j
) L ( A − 6)
Where, So,j = average liquid saturation at
location j, Sg,j = average gas saturation at
location j. Using the constrain equation So,j +
Sg,j = 1 and Eq. A-4, A-5 and A-6, So,j can be
solved as:
13
SCA 2001-35
( ) ( )
S o, j =
L (A − 7 )
ln(I C , j ) − ln(I S , j )
decreased 24. However, in order to reduce the
standard error, one cannot simply set the
counting time infinite rather it must be
optimized according to an accuracy one wish to
achieve from the measurement.
ln I C , j − ln I T , j
In order to obtain capillary pressure-saturation
function, average capillary pressure at location
j may be related to So,j since So,j represents
average liquid saturation at this location. The
average capillary pressure (Pc) at location j
may be obtained integrating the Pc over the
entire x-section at j and dividing it by the xsectional area at that location. The average Pc
at location j may then be calculated from the
following function22:
PC, j =
The process of setting the counting time for a
dependent random variable (for example So,j)
may be determined if system's standard error
could be described as a function time. If a
function, u can be formulated by a set of
independent random variables (x1…xk ) such
that u=u(x1…xk ) then the standard error for u
may be written as
1
3  4

∆ρ ω 2  r22 − r j2 + R 2  + ∆ρ g R L( A − 8)
2
4  3π

σ u2
k
(t ) = ∑
i =1
 ∂u

 ∂x
 i
2

 σ x2 (t )L ( A − 9 )

i

Where, σu(t) and σxi(t), i=1, k are standard
errors for u and xi which are functions of time t.
Note that if σxi(t) ≠ a function time t then σu(t)
= constant.
Where, PC,j = average Pc at location j, ∆ρ=
density difference between liquid and gas, ω=
centrifuge speed, g= acceleration of gravity,
R= sample radius, r2= distance between
rotation axis and center of the sample at outer
end or at gas-oil contact. rj= distance between
rotation axis and center of the sample at
location j. The Pc-saturation function may then
be obtained from the following data set {
So,j,PC,j j =1, N}
Now, we are interested to solve Eq.A-9 for So, j
so that the standard error associated with it can
be optimized. Replacing u = So,j and xi's by
ln(IC,j), ln(IS,j) and ln(IT,j), we can write
standard error for So,j approximately as22:
σ So, j ≈ α
The scanning interval length i.e. (r j+1-rj) may be
set for example 1mm in order for better
resolution. However, in order to set counting
time for j-locations, one must consider
statistical
uncertainties
involved
with
radioactive measurements.
1
t
L ( A − 10)
Where, σSo,j = standard error for So,j . α is
defined in the following equation and it may be
treated as constant for a given rock-fluid and γsource.
All radioactive counting follows Poissonian
statistics23. Therefore, standard error associated
to a radioactive measurement may be
calculated from the square root of the
magnitude of the measurement. If counting
time increased, the magnitude of the
counts/events also increased, as a result, the
standard error associated to an average quantity
derived from the radioactive measurement is
α=
3
I S , j ln
I C, j
L ( A − 11)
I S, j
Setting σSo,j = 0.01, the optimum counting time
t=20 minutes has been determined for our
system which has then been used at all
scanning locations i.e. j =1, N.
14
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